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Incorporating the Effect of Gravity Into Image‐Based Drainage Simulations on Volumetric Images of Porous Media

DOI:10.1029/2021WR031509
Authors:
Eric Alexander Chadwick at University of Toronto
Eric Alexander Chadwick
Lucas H. Hammen
Lucas H. Hammen
Lucas H. Hammen
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V. Paul Schulz at Duale Hochschule Baden-Württemberg Mannheim
V. Paul Schulz
Aimy Bazylak
Aimy Bazylak
Aimy Bazylak
  • This person is not on ResearchGate, or hasn't claimed this research yet.
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Abstract and Figures

Simulating drainage in volumetric images of porous materials is a key technique for studying multiphase flow and transport. Image‐based techniques based on sphere insertion are popular due to their computational efficiency and reasonable predictions, though they lack physical rigor. Since most tomograms are small, the impact of gravity on the fluid distributions has not been previously considered. With the advent of stochastically generated images of arbitrary size, and ever larger field‐of‐view images, the validity of neglecting gravity is becoming questionable. In this work, an image‐based technique that includes the effect of gravity during gravity stabilized displacements was developed and validated. Results compared favorably with analytical solutions of capillary rise in tubes, and to micromodel experiments in terms of the pseudo‐capillary pressure curves. The compactness of the invasion front was also shown to vary linearly with the inverse Bond number. Finally, a contour map of expected error as a function of image size and Bond number was generated to help identify when gravitational effects cannot be ignored. The presented algorithm utilizes only basic image processing tools and offers the same computational advantage as other image‐based sphere insertion methods.
A schematic describing the traditional full morphology method simulating drainage in porous media originating from the bottom of the image. The inlet is denoted by the red line in (c)–(e). (a) The starting image filled with the wetting phase. (b) The distance transform of (a) showing the minimum distance from each pixel of wetting phase to the nearest solid phase. (c) Seed points (dark gray) defined by thresholding the distance transform in (b). (d) Disconnected seed points removed. (e) Disks added to seed points (light gray).
A schematic describing the traditional full morphology method simulating drainage in porous media originating from the bottom of the image. The inlet is denoted by the red line in (c)–(e). (a) The starting image filled with the wetting phase. (b) The distance transform of (a) showing the minimum distance from each pixel of wetting phase to the nearest solid phase. (c) Seed points (dark gray) defined by thresholding the distance transform in (b). (d) Disconnected seed points removed. (e) Disks added to seed points (light gray).
… 
Schematic of seeding process and subsequent placing of disks using the proposed algorithm without gravity (a, c, e) and with gravity (b, d, f) for a fully nonwetting phase invading a fully wetting phase. (a) Regions are colored based on the corresponding capillary pressure at each pixel which is proportional to the distance transform. (b) Regions are colored based on the corresponding combined capillary and hydrostatic pressures at each pixel which is proportional to the distance transform and the height of each pixel. (c) Surviving pixels of the invading phase (dark gray) are placed inside at voxels where Pin≥Pc ${\boldsymbol{P}}_{\boldsymbol{in}}\ge {\boldsymbol{P}}_{\boldsymbol{c}}$. (d) Surviving pixels of the invading phase (dark gray) are placed inside at voxels where Pin≥Pc′ ${\boldsymbol{P}}_{\boldsymbol{in}}\ge {\boldsymbol{P}}_{\boldsymbol{c}}^{\prime }$. A close‐up is shown to highlight the metrics, r(x,y) obtained from the distance transform, and h(x,y), the height of the pixel along the y‐axis. (e)–(f) Insertion of disks at the surviving pixels using the distance transform.
Schematic of seeding process and subsequent placing of disks using the proposed algorithm without gravity (a, c, e) and with gravity (b, d, f) for a fully nonwetting phase invading a fully wetting phase. (a) Regions are colored based on the corresponding capillary pressure at each pixel which is proportional to the distance transform. (b) Regions are colored based on the corresponding combined capillary and hydrostatic pressures at each pixel which is proportional to the distance transform and the height of each pixel. (c) Surviving pixels of the invading phase (dark gray) are placed inside at voxels where Pin≥Pc ${\boldsymbol{P}}_{\boldsymbol{in}}\ge {\boldsymbol{P}}_{\boldsymbol{c}}$. (d) Surviving pixels of the invading phase (dark gray) are placed inside at voxels where Pin≥Pc′ ${\boldsymbol{P}}_{\boldsymbol{in}}\ge {\boldsymbol{P}}_{\boldsymbol{c}}^{\prime }$. A close‐up is shown to highlight the metrics, r(x,y) obtained from the distance transform, and h(x,y), the height of the pixel along the y‐axis. (e)–(f) Insertion of disks at the surviving pixels using the distance transform.
… 
Stochastically generated porous media images of (a) overlapping disks (2D) (b) overlapping spheres (3D), and (c) the non‐overlapping spheres (3D) of Ayaz et al. (2020). Solid pixels are shown in black in (a) and gray in (b) and (c) for increased visibility.
Stochastically generated porous media images of (a) overlapping disks (2D) (b) overlapping spheres (3D), and (c) the non‐overlapping spheres (3D) of Ayaz et al. (2020). Solid pixels are shown in black in (a) and gray in (b) and (c) for increased visibility.
… 
(a) Results of the IBSI with due regard to gravity in 10 capillary tubes from sizes 1 to 10 mm with a pixel resolution of Lvx=0.1mmpx ${\boldsymbol{L}}_{\boldsymbol{vx}}=0.1\frac{\mathbf{m}\mathbf{m}}{\mathbf{p}\mathbf{x}}$. (b) Meniscus height as a function of inlet pressure for IBSI simulations with gravity (solid markers) and theoretical calculations using Young‐Laplace equation (solid lines).
(a) Results of the IBSI with due regard to gravity in 10 capillary tubes from sizes 1 to 10 mm with a pixel resolution of Lvx=0.1mmpx ${\boldsymbol{L}}_{\boldsymbol{vx}}=0.1\frac{\mathbf{m}\mathbf{m}}{\mathbf{p}\mathbf{x}}$. (b) Meniscus height as a function of inlet pressure for IBSI simulations with gravity (solid markers) and theoretical calculations using Young‐Laplace equation (solid lines).
… 
Resolution study using capillary tubes of varying widths (1–1,000 mm) and image resolutions (Lvx=0.01–1mmpx ${\boldsymbol{L}}_{\boldsymbol{vx}}=0.01\,\,\mbox{--}\,\,1\,\frac{\boldsymbol{mm}}{\boldsymbol{px}}$). Mean absolute error of the meniscus height divided by capillary tube diameter in each tube is presented using a log scale to demonstrate the quickly decreasing error as the image resolution is increased.
+7
Resolution study using capillary tubes of varying widths (1–1,000 mm) and image resolutions (Lvx=0.01–1mmpx ${\boldsymbol{L}}_{\boldsymbol{vx}}=0.01\,\,\mbox{--}\,\,1\,\frac{\boldsymbol{mm}}{\boldsymbol{px}}$). Mean absolute error of the meniscus height divided by capillary tube diameter in each tube is presented using a log scale to demonstrate the quickly decreasing error as the image resolution is increased.
… 
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1. Introduction
Simulations of two-phase flow in porous media inform a wide range of applications in fields as diverse as ground-
water remediation, geologic storage of carbon dioxide, enhanced oil recovery, and electrochemistry (Hilpert
& Miller,2001; Schulz etal., 2007; Shikhov & Arns, 2015; Weishaupt etal., 2019). Continuum models are
widely used for studying two-phase flow (Amirebrahimi & Herrmann,1993; Celia etal.,1995,2015; Chaouche
etal.,1994) and are especially useful when macroscopic behavior is of interest such as for flow in deformable
porous media (Khoei etal.,2015) or electrochemical processes (Zenyuk etal.,2015). However, continuum models
offer limited insight into underlying physical phenomena occurring on the pore scale. When pore-scale informa-
tion about two-phase flow is of interest, there are several options including the Lattice-Boltzmann, volume-
of-fluid, and level-set methods, but these require a high amount of memory and computation times (Jettestuen
etal.,2013; Schulz etal.,2015). A more computationally feasible, though less rigorous alternative is to approx-
imate the fluid distributions at the pore-scale using image-analysis techniques. Morphological image opening
(MIO) and other image-based sphere insertion (IBSI) methods are well-established means of approximating fluid
distributions during quasistatic drainage in volumetric images of porous media (i.e., obtained from tomography
or serial-sectioning). Several review and research articles have demonstrated the reasonable effectiveness of this
approach by comparison to rigorous Lattice Boltzmann models (Pot etal.,2015; Vogel etal.,2005). The method
has also been shown to compare well with experimental data in terms of observed capillary pressure curves
(Sweijen etal.,2017; Thakur etal.,2019). So, although approximate, the method is quite useful and worthy of
extending to more diverse applications. IBSI (sometimes known as the full morphology method or FMM) was
first introduced by Hazlett(1995) and further developed to operate using morphological image operations by
Hilpert and Miller(2001). It has since been improved and benchmarked against other methods for simulating
flows in porous media (V. P. Schulz etal., 2015; Vogel etal., 2005). In all IBSI implementations so far, the
effects of gravity have been neglected. This omission is normally valid for many applications which are on small
enough length scales that the impact of gravity on fluid distribution is negligible. This is usually true on the
scale of most X-ray tomography images for instance. However, in applications with larger length scales, such
as sand column experiments, vertical micromodels, and large-scale electrochemical devices, gravity may play a
Abstract Simulating drainage in volumetric images of porous materials is a key technique for studying
multiphase flow and transport. Image-based techniques based on sphere insertion are popular due to their
computational efficiency and reasonable predictions, though they lack physical rigor. Since most tomograms
are small, the impact of gravity on the fluid distributions has not been previously considered. With the advent
of stochastically generated images of arbitrary size, and ever larger field-of-view images, the validity of
neglecting gravity is becoming questionable. In this work, an image-based technique that includes the effect of
gravity during gravity stabilized displacements was developed and validated. Results compared favorably with
analytical solutions of capillary rise in tubes, and to micromodel experiments in terms of the pseudo-capillary
pressure curves. The compactness of the invasion front was also shown to vary linearly with the inverse Bond
number. Finally, a contour map of expected error as a function of image size and Bond number was generated
to help identify when gravitational effects cannot be ignored. The presented algorithm utilizes only basic image
processing tools and offers the same computational advantage as other image-based sphere insertion methods.
CHADWICK ET AL.
© 2022. American Geophysical Union.
All Rights Reserved.
Incorporating the Effect of Gravity Into Image-Based
Drainage Simulations on Volumetric Images of Porous Media
Eric A. Chadwick1,2, Lucas H. Hammen2, Volker P. Schulz2 , Aimy Bazylak1 ,
Marios A. Ioannidis3 , and Jeff T. Gostick3
1Thermofluids for Energy and Advanced Material Laboratory, Department of Mechanical and Industrial Engineering,
Institute for Sustainable Energy, Faculty of Applied Science and Engineering, University of Toronto, Toronto, ON, Canada,
2Department of Mechanical Engineering, Electrochemistry Research Cluster, Duale Hochschule Baden-Württemberg,
Mannheim, Germany, 3Porous Materials Engineering and Analysis Lab, Department of Chemical Engineering, University of
Waterloo, Waterloo, ON, Canada
Key Points:
• The effect of gravity was incorporated
in an image-based drainage algorithm
• Comparison to micromodel
experiments from literature
showed excellent agreement in
pseudo-capillary pressure curves,
quantitatively predicting the slope
while qualitatively matching the
residual saturation
• Expected error for neglecting gravity
was compiled into a contour plot to
estimate which images and fluid pairs
should consider gravity, with errors as
high as 50% in some cases
Correspondence to:
J. T. Gostick,
jgostick@uwaterloo.ca
Citation:
Chadwick, E. A., Hammen, L. H., Schulz,
V. P., Bazylak, A., Ioannidis, M. A.,
& Gostick, J. T. (2022). Incorporating
the effect of gravity into image-based
drainage simulations on volumetric
images of porous media. Water Resources
Research, 58, e2021WR031509. https://
doi.org/10.1029/2021WR031509
Received 29 OCT 2021
Accepted 31 JAN 2022
Author Contributions:
Conceptualization: Jeff T. Gostick
Formal analysis: Lucas H. Hammen,
Aimy Bazylak, Marios A. Ioannidis
Funding acquisition: Aimy Bazylak
Investigation: Volker P. Schulz, Marios
A. Ioannidis
Methodology: Lucas H. Hammen, Jeff
T. Gostick
Project Administration: Volker P.
Schulz, Aimy Bazylak
Resources: Volker P. Schulz, Aimy
Bazylak
Software: Eric A. Chadwick, Lucas H.
Hammen, Jeff T. Gostick
Supervision: Volker P. Schulz, Aimy
Bazylak
Validation: Eric A. Chadwick, Lucas H.
Hammen, Volker P. Schulz, Marios A.
Ioannidis, Jeff T. Gostick
Visualization: Eric A. Chadwick
Writing – original draft: Eric A.
Chadwick, Volker P. Schulz, Aimy
10.1029/2021WR031509
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